3) STUDIES ON FINITE STRAIN AND STRESS WITH MATRIX METHOD : The Basic Study on Finite Deformation of Structures, 1

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  • 有限歪および応力の三次元的取扱いに就いて : 構造物の有限変形に関する基礎研究・その1
  • 有限歪および応力の三次元的取扱いに就いて
  • ユウゲン ワイ オヨビ オウリョク ノ 3ジゲンテキ トリアツカイ ニ ツイテ

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When a medium be acted on by deforming forces, the position of the medium before and after deformation will be called the initial and final stats of the medium respectively. Let a^i (a^1a^2a^3) be the rectangular cartisian coordinate of a representative particle of the medium in the initial state, and x^i(x^1x^2x^3) be the rectangular cartisian coordinate of the corresponding particle in the final state. Then the elastic deformation is represented by particle to particle transformations. a^i→x^i=f^i(a) Hence, with matrices representation a→x=Ra The clasical theory of elastic bodies asoumes that the deformation eq. (1) are "infinitesimol". But such approximation is not sufficient in some study on slender rods, long collumns and thin shells and plates etc. As a result the finite deformation theory be used in enginering problemes. This paper is conceined with the matrix repiesentative of non-linear strain for finite strain with the F.D. Murnaghan's and A.D. Michal's equation. Theas strains are calculated the following deformation states, (1) Simple elongation of rod, (2) Isotropic elongation of plate, (3) Plane shear defformation (4) Sheardeformation of space, (5) Deformation by isotropic compression, (6) Torsion of rod, (7) Simple bending. In the next section, be given a various matrix form of stress tensor, and calculate linear or non-linear stresses corresponding the various deformations. Then theas stresses from deformations were contained (1) Shearing stress, (2) Tensing stress, (3) Bending stress (4) Stress at large deflected plate. Finally, this finite strain ε_f be compared with the logarithmic strain ε^^- and linear strain ε on a cartisian coordinate, which consist of xaxis (l/l_0) and y axis (ε, ε^^- and ε_f).

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